Microcausality without Lorentz invariance

TL;DR

This paper explores how the principle of microcausality applies in Lorentz-breaking quantum field theories, deriving conditions on two-point functions and implications for effective field theories in cosmological and condensed matter contexts.

Contribution

It extends the concept of microcausality to Lorentz-breaking states and derives analyticity and positivity conditions applicable to low-energy effective theories.

Findings

Derived analyticity and exponential boundedness conditions for two-point functions.

Identified a positivity condition on an EFT coefficient during inflation.

Discussed how microcausality constrains higher-point correlation functions.

Abstract

Microcausality -- the vanishing of commutators outside the lightcone -- is a fundamental property of relativistic quantum field theories. We derive its implications for two-point functions of scalar operators on {\it Lorentz-breaking} states. We restrict to spatially homogeneous and isotropic states, at zero and finite temperature, such as finite-density states of matter and primordial inflationary states. In a mixed $(t, \vec k \, )$ representation, we find certain analyticity and exponential boundedness conditions, which we verify in a variety of examples. Crucially, we discuss how our conditions can be tested within the regime of validity of Lorentz-breaking low-energy effective field theories, clarifying the role of the group velocity of low-energy excitations. In the cosmological case, we derive a positivity condition on an EFT coefficient in an inflationary background. Lastly, we comment on how microcausality can be used to constrain higher-point correlation functions, via suitable nested commutators.